#72322325

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24128
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2545 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(6,12,14,28)(7,16)(8,10)(9,22)(13,29,17,35)(15,33,26,23)(18,19)(21,25)(27,32,31,30)(34,36);(5,6,8,15,11,14,10,26)(7,17,19,32,18,13,16,30)(9,23,21,28,25,33,22,12)(20,35,36,27,24,29,34,31);(1,2)(3,4)(7,18)(16,19)(20,24)(34,36);(1,3,4,2)(5,7,10,19,11,18,8,16)(6,13,26,30,14,17,15,32)(9,24,22,34,25,20,21,36)(12,27,33,29,28,31,23,35)

Polycyclic presentation:

2^10|a^2=e,b^2=f,d^2=efgij,f^2=j,g^2=j,h^2=i,i^2=j,b^a=bh,c^a=cfgj,c^b=cefi,d^a=dfgi,d^b=de,d^c=defg,e^b=ej,f^a=fj,f^d=fj,g^a=gj,g^d=gj,h^a=hi,h^b=hi,h^d=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322329, #72322804, #72322868, …
maximal normal subgroup of	#72322329, #72322804, #72322868, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 4, F.5*F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);